Q:

# Kim has a strong first​ serve; whenever it is good​ (that is,​ in) she wins the point 70​% of the time. Whenever her second serve is​ good, she wins the point 40​% of the time. Fifty dash five percent of her first serves and 70​% of her second serves are good. ​(a) What is the probability that Kim wins the point when she​ serves? ​(b) If Kim wins a service​ point, what is the probability that her 1st serve was​ good?

Accepted Solution

A:
Answer:a) There is a 63.35% probability that Kim wins the point when she serves.b) If Kim wins a service​ point, there is a 55.80% probability that her 1st serve was​ good.Step-by-step explanation:We have these following probabilitiesA 50.5% probability that her first serve is good.A 70% probability that her second serve is good.If her first serve is good, she has a 70% probability of winning the point.If her second serve is good, a 40% probability of winning the point.​(a) What is the probability that Kim wins the point when she​ serves?This is the sum of 70% of 50.5% and 40% of 70%. So$$P = 0.7*(0.505) + 0.4*(0.7) = 0.6335$$There is a 63.35% probability that Kim wins the point when she serves.​(b) If Kim wins a service​ point, what is the probability that her 1st serve was​ good?This can be formulated as the following problem:What is the probability of B happening, knowing that A has happened.It can be calculated by the following formula$$P = \frac{P(B).P(A/B)}{P(A)}$$Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.SoWhat is the probability of Kim's first serve being good, given that she won the point?P(B) is the probability of Kim's first serve being good. So $$P(B) = 0.505$$.P(A/B) is the probability of Kim's winning the point when her first serve is good. So $$P(A/B) = 0.70$$.P(A) is the probability of Kim's winning the point. From a), that is $$P(A) = 0.6335$$$$P = \frac{P(B).P(A/B)}{P(A)} = P = \frac{0.505*0.70}{0.6335} = 0.5580$$If Kim wins a service​ point, there is a 55.80% probability that her 1st serve was​ good.